Bounds on antipodal spherical designs with few angles

Abstract

A finite subset $X$ on the unit sphere ${\mathbb{S}}^{d-1}$ is called an $s$-distance set with strength $t$ if its angle set $A(X):=\{⟨\mathbf{x},\mathbf{y}⟩:\mathbf{x},\mathbf{y}\in X,\mathbf{x}\ne \mathbf{y}\}$ has size $s$, and $X$ is a spherical $t$-design but not a spherical $(t+1)$-design. In this paper, we consider to estimate the maximum size of such antipodal set for small $s$. First, we improve the known bound on $|X|$ for each even integer $s\in [\frac{t+5}{2},t+1]$ when $t\ge 3$. We next focus on two special cases: $s=3,t=3$ and $s=4,t=5$. Estimating the size of $X$ for these two cases is equivalent to estimating the size of real equiangular tight frames (ETFs) and Levenstein-equality packings, respectively. We first improve the previous estimate on the size of real ETFs and Levenstein-equality packings. This in turn gives a bound on $|X|$ when $s=3,t=3$ and $s=4,t=5$, respectively.

20 pages